On the polar derivative of lacunary type polynomials

Year 2024, , 1197 - 1209, 30.12.2024

Fatemeh Mohammadi , Ahmad Motamednezhad

https://doi.org/10.31801/cfsuasmas.1521079

Abstract

Let $p(z)=a_nz^n+\sum_{l=\nu}^na_{n-l}z^{n-l}$, where $1\leq \nu \leq n$, be a polynomial of degree $n$ having all its zeros in $|z|\leq k\leq 1$. For polar derivative $D_{\alpha}p(z)$, it is known that for each $|\alpha|\leq 1$ on $|z|=1$,
\begin{align*}
|D_{\alpha}p(z)|\leq \frac{n}{1+k^{\nu}}\Big\{(|\alpha|+k^{\nu})\|p\|_{\infty}-\frac{1-|\alpha|}{k^{n-\nu}}\min_{|z|=k}|p(z)|\Big\}.
\end{align*}
In this paper, we obtain the $L_q$ mean extension and a refinement of the above and other related results for the polar derivative of polynomials.

Keywords

Derivative, lacunary type polynomial, $L^q$ Inequality, maximum modulus, restricted zeros

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